E(X_t) 구하기
E(X_t) = E(\lambda X_{t-1}) =
\lambda~E(X_{t-1}) = \lambda^t~E(X_0)
Var(X_t) 구하기
Var(X_t) =
E\big(Var(X_t|X_{t-1})\big) +
Var\big(E(X_t|X_{t-1})\big)
1. Var(E(X_t|X_{t-1})) 구하기
\begin{align*}
&E(X_t |X_{t-1} = x) =
E(Y_{t-1,~1} + \cdots + Y_{t-1,~n}
~|~X_{t-1}=x)
\\[10pt]
&= E(Y_{t-1,~1}~|~X_{t-1}=x) +
\cdots+E(Y_{t-1,~n}~|~X_{t-1}=x)
\\[10pt]
&= E(Y_{t-1,~1}) + \cdots +
E(Y_{t-1,~n})
\\[10pt]
&= \lambda \cdot x
\\[25pt]
&\therefore~E(X_t~|~X_{t-1}) = \lambda \cdot X_{t-1}
\end{align*}
Var\big(E(X_t~|~X_{t-1})\big) =
Var(\lambda X_{t-1}) =
\lambda^2~Var(X_{t-1})
2. E(var(X_t|X_{t-1})) 구하기
\begin{align*}
Var(X_t~|~X_{t-1}=x) &=
Var(Y_{t-1,~1} + \cdots +
Y_{t-1,~n}~|~X_{t-1}=x)
\\[10pt]
&=Var(Y_{t-1,~1} + \cdots +
Y_{t-1,~n})
\\[10pt]
&= \lambda \cdot x
\\[20pt]
Var(X_t~|~X_{t-1}) &=
\lambda \cdot X_{t-1}
\end{align*}
\begin{align*}
E\big(Var(X_t~|~X_{t-1})\big) &=
E(\lambda \cdot X_{t-1})
= \lambda \cdot E(X_{t-1})
\\[10pt]
&=\lambda^2 \cdot E(X_{t-2})
\\[10pt]
&=\lambda^t \cdot E(X_{0})
\end{align*}
3. Var(X_t) 구하기
\begin{align*}
Var(X_t) &=
E\big(Var(X_t|X_{t-1})\big) +
Var\big(E(X_t|X_{t-1})\big)
\\[10pt]
&= \lambda^t \cdot E(X_{0}) +
\lambda^2~Var(X_{t-1})
\\[20pt]
\therefore Var(X_t)
&= \lambda^t \cdot E(X_{0}) +
\lambda^2~Var(X_{t-1})
\end{align*}
4. Var(X_t)의 점화식 구하기
위에서 구한 식은 계산과정이 복잡하다는 단점이 있다.
점화식을 구해 이를 해결해보자.
Var(X_1) - \lambda^2Var(X_0) = \lambda E(X_0)
\\
Var(X_2) - \lambda^2Var(X_1) = \lambda^2 E(X_0)
\\
Var(X_3) - \lambda^2Var(X_2) = \lambda^3 E(X_0)
\\
Var(X_4) - \lambda^2Var(X_3) = \lambda^4 E(X_0)
\\
\vdots
\\
Var(X_t) - \lambda^2Var(X_{t-1}) = \lambda^t E(X_0)
\lambda^2Var(X_1) - \lambda^4 Var(X_0) = \lambda^3E(X_0)
\\
Var(X_2) - \lambda^2 Var(X_1) = \lambda^2E(X_0)
\\[20pt]
Var(X_2) - \lambda^4 Var(X_0) = E(X_0)\cdot(\lambda^2 + \lambda^3)
\\
Var(X_2) = E(X_0)
\cdot(\lambda^2 + \lambda^3)+
\lambda^4 Var(X_0)
\\
Var(X_0) -\lambda^2 \Big(E(X_0)(\lambda^2+\lambda^3) + \lambda^4
E(X_0)\Big) = \lambda^3E(X_0)
\\[10pt]
Var(X_0) = E(X_0)(\lambda^4+\lambda^5)
+ \lambda^3 E(X_0) + \lambda^6 Var(X_6)
\\=E(X_0)(\lambda^3+\lambda^4+\lambda^5) + \lambda^6Var(X_0)
Var(X_4) - \lambda^2\big(E(X_0)(\lambda^3+\lambda^4+\lambda^5)
+ \lambda^6Var(X_0)
\big)
=\lambda^4E(X_0)
\\[15pt]
Var(X_4)- E(X_0)
(\lambda^5+\lambda^6+\lambda^7) + \lambda^8Var(X_0)
=\lambda^4E(X_0)
\\[15pt]
Var(X_4) = E(X_0)
(\lambda^4+\lambda^5+\lambda^6+\lambda^7) + \lambda^8Var(X_0)
\\
\vdots
\\
Var(X_n) = E(X_0)
(\lambda^n+\cdots+\lambda^{2n-1}) + \lambda^{2n}Var(X_0)
Var(X_n) = E(X_0) \cdot
{\lambda^n(\lambda^n-1) \over\lambda-1} +
\lambda^{2n}~Var(X_0)